series convergence evaluation

[sambellamy]

Hello. I am working with a problem where we're asked to determine whether the series in convergent or divergent, and if it is convergent, to find what it converges to:

Σ_n=1^∞ (3/5ⁿ + 2/n)

I have determined that the 3/5ⁿ portion is a geometric series, with a=3 and r=1/5, and the limit of the partial sums is 15/4. I am not sure how to deal with 2/n however. I understand it is not simply the limit of 2/n, which I know to be zero, but the sum of 2/n as n increases without bound (2+1+2/3,+...). should I also categorize this as a geometric series, and how so?

Thanks in advance.

 

[pka]

Hello. I am working with a problem where we're asked to determine whether the series in convergent or divergent, and if it is convergent, to find what it converges to:
Σ_n=1^∞ (3/5ⁿ + 2/n)
I have determined that the 3/5ⁿ portion is a geometric series, with a=3 and r=1/5, and the limit of the partial sums is 15/4. CORRECT.

However, the series \(\displaystyle \displaystyle\sum\limits_{n = 1}^\infty {\frac{2}{n}} \) diverges. SO?

 

[Ishuda]

To expand a little on the pka post. Have you heard of the harmonic series, the series whose terms are 1, 1/2, 1/3, 1/4, etc, and the fact that it diverges? If a series diverges, then a constant times that series also diverges.

 

[HallsofIvy]

If \(\displaystyle \sum a_n\) and \(\displaystyle \sum b_n\) both converge then so does \(\displaystyle \sum (a_n+ b_n)\) because \(\displaystyle \sum (a_n+ b_n)= \sum a_n+ \sum b_n\). One consequence of that is that if \(\displaystyle \sum a_n\) converges and \(\displaystyle \sum b_n\) does not then \(\displaystyle \sum (a_n+ b_n)\) cannot converge. If it did then so would \(\displaystyle \sum [(a_n+ b_n)+ (-a_n)]\).

 

[sambellamy]

excellent feedback, thank you!